Alòs type decomposition formula for Barndorff-Nielsen and Shephard model
AAl`os type decomposition formula for Barndorff-Nielsen andShephard model
Takuji Arai ∗ May 18, 2020
Abstract
The objective is to provide an Al`os type decomposition formula and an approximate optionpricing formula for the Barndorff-Nielsen and Shephard model: an Ornstein-Uhlenbeck typestochastic volatility model driven by a subordinator without drift. Al`os [2] introduced a de-composition expression of the call option prices for the Heston model by using Ito’s formula. Inthis paper, we extend it to the Barndorff-Nielsen and Shephard model. As far as we know, thisis the first result on the Al`os type decomposition formula for models with infinite active jumps.Moreover, investigating the rate of convergence as the time to maturity tends to 0 for eachterm in the obtained decomposition formula, we shall present an approximate option pricingformula, and implement numerical experiments, which show that our approximation formulais effective for in-the-money options.
Stochastic volatility models have drawn considerable attention in mathematical finance since theyare very useful for capturing the volatility skew and smiles, but there is no closed-form optionpricing formula for stochastic volatility models in general. Thus, some authors have presenteddecomposition expressions of option prices, which are useful to derive approximations of optionprices and to analyze implied volatilities. Firstly, for continuous stochastic volatility models withno correlation between the asset price and the volatility processes, Hull and White [13] providedan option price expression with a conditional expectation of the Black-Scholes formula by substi-tuting the future average volatility for the volatility in the Black-Scholes formula. Al`os [1] hasextended it to correlated models by means of Malliavin calculus in order to deal with Ito’s formulafor anticipating processes, since the future average volatility is a non-adapted process. Besides,extensions to more general models have been done by [4], [5], [12] and so on. On the other hand,Al`os [2] obtained a new decomposition formula for the Heston model by using the average squaredfuture volatility, instead of the future average volatility. Since the average squared future volatilityis an adapted process, she made use of the classical Ito calculus, not the Malliavin calculus. Thedecomposition formula in [2] is given as the sum of the Black-Scholes formula and terms due to thevolatility process. In addition, using the obtained decomposition expression, approximate option ∗ Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan.([email protected]) a r X i v : . [ q -f i n . M F ] M a y ricing formulas were also presented. This Al`os type decomposition formula has been extendedto more general models by [15], [16] and so on. Among them, Merino et al. [14] has extended tostochastic volatility models with finite active jumps. Moreover, for the Heston model, Al`os et al. [3]suggested an approximation of the implied volatility and a calibration method by using the resultsof [2].The objective of this paper is to obtain an Al`os type decomposition formula and an approx-imate option pricing formula for the Barndorff-Nielsen and Shephard (BNS) model. To our bestknowledge, this is the first result of the Al`os type decomposition formula for models with infiniteactive jumps, but Jafari and Vives [12] derived a Hull-White type decomposition formula for modelswith infinite active jumps by means of Mailliavin calculus. Now, the BNS model is a representativejump-type stochastic volatility model undertaken by [9], [10], and its volatility process is given bya non-Gaussian Ornstein-Uhlenbeck process. For details on the BNS model, see also [17] and [18].The BNS model has the following three features: First, the asset price process has jumps, but alljumps are negative. Second, there is no Brownian component in the volatility process. Third, thejump component is common between the asset price and the volatility processes. Remark that thejumps might be infinite active. Main results of this paper will be derived by making the most ofthese features of the BNS model.To introduce our results in detail, we obtain a decomposition expression for the vanilla calloption prices by applying Ito’s formula to the Black-Scholes formula. It is given as the sum of theBlack-Scholes formula, a term due to the impact of the asset price jumps, and some residual termsdue to the asset price jumps and changes of the volatility. Unlike [2], we use the current squaredvolatility value instead of the average squared future volatility, and substitute it to the volatility inthe Black-Scholes formula.In addition to this, we present a numerically tractable approximation formula by investigatingthe rate of convergence as the time to maturity tends to 0 for each term in our decompositionformula. Note that our approximation formula is given as the sum of the Black-Scholes formulaand a correction term corresponding to the possibility of that the payoff of an in-the-money(ITM)option may vanish in a moment due to a big jump of the asset price. Thus, the correction termbecomes 0 if the option is out-of-the-money (OTM). Besides, we also implement some numericalexperiments in order to make sure how effective our approximation formula is for ITM options.The structure of this paper is as follows: We give some mathematical preliminaries and notationsin the following section. Section 3 introduces our main theorems: decomposition and approximationformulas. Subsection 3.3 is devoted to numerical results. Proofs of the main theorems are given inSections 4 and 5, respectively. Conclusions are summarized in Section 6. Consider throughout a financial market model in which only one risky asset and one riskless assetare tradable. Let r ≥ T > t ∈ [0 , T ] is described by S t := S exp (cid:26)(cid:90) t (cid:18) r + µ −
12 Σ u (cid:19) du + (cid:90) t Σ u dW u + ρH λt (cid:27) , t ∈ [0 , T ] , (2.1)2here S > ρ ≤ µ ∈ R , λ > H is a subordinator without drift, and W is a 1-dimensionalstandard Brownian motion. Here Σ is the volatility process, of which squared process Σ is given byan Ornstein-Uhlenbeck process driven by the subordinator H λ , that is, the solution to the followingstochastic differential equation: d Σ t = − λ Σ t dt + dH λt , t ∈ [0 , T ] (2.2)with Σ >
0. Note that the asset price process S is defined on some filtered probability space(Ω , F , ( F t ) ≤ t ≤ T , P ) with the usual condition, where ( F t ) ≤ t ≤ T is the filtration generated by W and H λ . In addition, we denote by X the log price process log S , that is, X t := log S t = log S + (cid:90) t (cid:18) r + µ −
12 Σ u (cid:19) du + (cid:90) t Σ u dW u + ρH λt , t ∈ [0 , T ] . (2.3)Remark that the term ρH λt in (2.3) (or (2.1)) accounts for the leverage effect, which is a stylizedfact such that the asset price declines at the moment when the volatility increases.For later use, we enumerate some properties of Σ: Firstly, we haveΣ t = e − λt Σ + (cid:90) t e − λ ( t − u ) dH λu ≥ e − λT Σ (2.4)for any t ∈ [0 , T ], that is, Σ is bounded from below. Next, the integrated squared volatility isrepresented as (cid:90) Tt Σ u du = (cid:15) ( T − t )Σ t + (cid:90) Tt (cid:15) ( T − u ) dH λu (2.5)for any t ∈ [0 , T ], and (cid:90) T Σ u du ≤ λ ( H λT + Σ ) , (2.6)where (cid:15) ( t ) := 1 − e − λt λ . Now, we denote by N the Poisson random measure of H λ . Hence, we have H λt = (cid:90) ∞ zN ([0 , t ] , dz ) , t ∈ [0 , T ] . Letting ν be the L´evy measure of H λ , we find that (cid:101) N ( dt, dz ) := N ( dt, dz ) − ν ( dz ) dt is the compensated Poisson random measure. Note that ν is a σ -finite measure on (0 , ∞ ) satisfying (cid:90) ∞ ( z ∧ ν ( dz ) < ∞ by Proposition 3.10 of [11]. The asset price process S is also given as the solution to the followingstochastic differential equation: dS t = S t − (cid:26) αdt + Σ t dW t + (cid:90) ∞ ( e ρz − (cid:101) N ( dt, dz ) (cid:27) , t ∈ [0 , T ] , α := r + µ + (cid:90) ∞ ( e ρz − ν ( dz ) . Note that S t > t ∈ [0 , T ].Now, we introduce our standing assumption as follows: Assumption 2.1. µ = (cid:90) ∞ (1 − e ρz ) ν ( dz ) .2. (cid:90) ∞ e (cid:15) ( T ) z ν ( dz ) < ∞ . The above condition 1 implies that the discounted asset price process (cid:98) S t := e − rt S t becomes a localmartingale. On the other hand, the condition 2 ensures that (cid:90) ∞ z ν ( dz ) < ∞ , which yields E [ H λT ] < ∞ by Proposition 3.13 of [11], and E (cid:34) sup t ∈ [0 ,T ] X t (cid:35) < ∞ (2.7)by (2.6). In addition, E (cid:34) sup t ∈ [0 ,T ] S t (cid:35) < ∞ (2.8)holds under the condition 2 from the view of Subsection 2.3 of [8]. Thus, (cid:98) S is a square-integrablemartingale under Assumption 2.1. Example 2.2.
We introduce two important examples of the squared volatility process Σ .1. The first one is the case where Σ follows an IG-OU process. The corresponding L´evy measure ν is given by ν ( dz ) = λa √ π z − (1 + b z ) exp (cid:26) − b z (cid:27) dz, z ∈ (0 , ∞ ) , where a > and b > . Note that this is a representative example of the BNS model withinfinite active jumps, that is, ν ((0 , ∞ )) = ∞ . In this case, the invariant distribution of Σ follows an inverse-Gaussian distribution with parameters a > and b > . Note that thecondition 2 of Assumption 2.1 is satisfied whenever b > (cid:15) ( T )
2. The second example is the gamma-OU case. In this case, ν is described as ν ( dz ) = λabe − bz dz, z ∈ (0 , ∞ ) , and the invariant distribution of Σ is given by a gamma distribution with parameters a > and b > . If b > (cid:15) ( T ) , then the condition 2 of Assumption 2.1 is satisfied. .2 Black-Scholes formula In this subsection, consider the so-called Black-Scholes model with volatility σ > r ≥
0, and the call option with strike price
K >
T >
0. We describe thecall option price at time t ∈ [0 , T ) with the log asset price x ∈ R by a function BS on not only t and x , but also squared volatility σ . Thus, the function BS ( t, x, σ ), which is well-known as theBlack-Scholes formula, is given as BS ( t, x, σ ) := e x Φ( d + ) − Ke − rτ t Φ( d − ) , t ∈ [0 , T ) , x ∈ R , σ > , (2.9)where τ t = T − t , Φ is the cumulative distribution function of the standard normal distribution,and d ± := x − log K + rτ t σ √ τ t ± σ √ τ t . (2.10)For later use, we denote x z := x + ρz, σ z := (cid:112) σ + z, η ± := r ± σ , η ± z := r ± σ z η ± ± z z > x ∈ R and σ >
0. Thus, d ± is rewritten as d ± = x − log K + η ± τ t σ √ τ t . Furthermore, we define d ± ρz := x z − log K + η ± τ t σ √ τ t = d ± + ρzσ √ τ t . (2.12)and d ± ρz,z := x z − log K + η ± z τ t σ z √ τ t (2.13)for z >
0. Remark that the time parameter t included in d ± , d ± ρz and d ± ρz,z might be replaced with u or s according to the situation. In addition, since we havelim t → T BS ( t, x, σ ) = ( e x − K ) + , the domain of the function BS can be extended to [0 , T ] × R × (0 , ∞ ), and we may define BS ( T, x, σ ) := ( e x − K ) + . For simplicity, substituting X t and Σ t defined in (2.3) and (2.2) for x and σ respectively in thefunction BS , we denote BS t := BS ( t, X t , Σ t )for t ∈ [0 , T ].More importantly, defining an operator D BS as D BS f ( t, x, σ ) := (cid:18) ∂ t + σ ∂ x + η − ∂ x − r (cid:19) f ( t, x, σ )5or R -valued function f ( t, x, σ ), t ∈ [0 , T ), x ∈ R , σ >
0, we have D BS BS ( t, x, σ ) = 0 , t ∈ [0 , T ) , x ∈ R , σ > . (2.14)Remark that partial derivatives of BS are given as ∂ x BS ( t, x, σ ) = e x Φ( d + ) , (2.15) ∂ x BS ( t, x, σ ) = e x Φ( d + ) + e x σ √ τ t φ ( d + ) , (2.16)and ∂ σ BS ( t, x, σ ) = τ t ∂ x − ∂ x ) BS ( t, x, σ ) = √ τ t σ e x φ ( d + ) , (2.17)where φ is the probability density function of the standard normal distribution. All of the abovederivatives are positive functions. For later use, we define additionally the following operators for R -valued function f ( t, x, σ ), t ∈ [0 , T ), x ∈ R , σ > a,b f ( t, x, σ ) := f ( t, x + a, σ + b ) − f ( t, x, σ ) , a, b ∈ R , L z f ( t, x, σ ) := ∆ ρz, f ( t, x, σ ) + ∂ x f ( t, x, σ )(1 − e ρz ) , z > , and L f ( t, x, σ ) := (cid:90) ∞ L z f ( t, x, σ ) ν ( dz ) . In this section, we introduce our main results: a decomposition formula and an approximationformula for the BNS model introduced in Section 2. Recall that the discounted asset price process (cid:98) S is a square-integrable martingale under Assumption 2.1. Thus, for the vanilla call option withstrike price K >
T >
0, its price at time t ∈ [0 , T ] is given as V t := e − rτ t E [ BS T | X t , Σ t ] . In Theorem 3.1 below, we derive a decomposition expression of V t by applying Ito’s formula tothe Black-Scholes function BS . Moreover, we provide in Theorem 3.6 an approximation of V t ,which is based on Theorem 3.1. Proofs of Theorems 3.1 and 3.6 will be given in Sections 4 and 5,respectively. Theorem 3.1.
Under Assumption 2.1, we have, for t ∈ [0 , T ] , V t = BS t + τ t L BS t + I + I + I + I + I . (3.1) Here, I , . . . , I are defined as follows: I := E (cid:34)(cid:90) Tt e − r ( u − t ) ∂ σ BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) , := E (cid:34)(cid:90) Tt e − r ( u − t ) (cid:90) ∞ (cid:0) ∆ ρz,z − ∆ ρz, (cid:1) BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) ,I := E (cid:34)(cid:90) Tt e − r ( u − t ) τ u ∂ x L BS u µdu (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) ,I := E (cid:34)(cid:90) Tt e − r ( u − t ) τ u ∂ σ L BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) , and I := E (cid:34) (cid:90) Tt e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) , where τ u := T − u . Remark 3.2.
In the decomposition formula (3.1), the first two terms in the right hand side areregarded as principal terms. In particular, the second term τ t L BS t represents the impact of thejumps of the asset price process. Indeed, it becomes 0 whenever ρ = 0 . Note that this term convergesto 0 with order 1 as the time to maturity τ t tends to 0. On the other hand, as seen in Section 5,the residual terms I , . . . , I converge to 0 with higher order than 1 as τ t tends to 0. Here we giveinterpretations of I , . . . , I in turn. First of all, we can say that I represents the influence of thecontinuous fluctuation of the squared volatility process Σ . Next, decomposing I into the followingtwo terms E (cid:34)(cid:90) Tt e − r ( u − t ) (cid:90) ∞ ∆ ,z BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) , (3.2) and E (cid:34)(cid:90) Tt e − r ( u − t ) (cid:90) ∞ (cid:0) ∆ ρz,z − ∆ ρz, − ∆ ,z (cid:1) BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) , (3.3) we can say that (3.2) represents the impact of the jumps of the squared volatility process, but (3.3)is corresponding to the impact of that jumps occur simultaneously in the asset price process and thesquared volatility process. As for the last three terms, the comparison between (3.1) and (4.4) belowgives I + I + I = E (cid:34)(cid:90) Tt e − r ( u − t ) L BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) − τ t L BS t . Thus, the sum I + I + I is corresponding to the residual part of the impact of the asset pricejumps. Each I , I and I represents the interaction of the impact of the asset price jumps with thecontinuous fluctuation of the asset price process, the continuous fluctuation of the squared volatilityprocess, and the fact that jumps occur simultaneously in the asset price and the squared volatilityprocesses, respectively. Remark 3.3.
As mentioned in Section 1, the decomposition formula (3.1) is given as an extensionof the result of [2] for Heston model, in which the average squared future volatility V t has beensubstituted for the volatility in the Black-Scholes formula, where V t is defined as V t := 1 τ t (cid:90) Tt E [Σ u | Σ t ] du. ote that V t for the BNS model is given as V t = (cid:15) ( τ t ) τ t Σ t + 1 λ (cid:18) − (cid:15) ( τ t ) τ t (cid:19) (cid:90) ∞ zν ( dz ) by (2.4). In this paper, we use the current squared volatility value Σ t , not V t , since the use of Σ t simplifies our calculations drastically. In addition, as indicated in Figure 2 below, the differencebetween BS t = BS ( t, X t , Σ t ) and BS ( t, X t , V t ) is sufficiently small. Thus, the choice of Σ t or V t does not make a big impact on the effectiveness of the approximation formula in Theorem 3.6. Using the decomposition formula (3.1), we present an approximate option pricing formula which isnumerically tractable. However, it is not easy to develop numerical methods for the terms I , . . . , I and τ t L BS t in (3.1). Thus, we need to exclude or replace these terms from our approximationformula. To this end, we shall show that I , . . . , I converge to 0 as the time to maturity goes to0 with higher order than 1. This fact allows us to exclude I , . . . , I from our approximation. Inaddition, we shall see that τ t L BS t converges as τ t → e − rτ t τ t (cid:90) ∞ z t ( K − e X t + ρz ) + { X t > log K } ν ( dz ) (3.4)for any z t ∈ [0 , τ t ]. As mentioned in Subsection 3.3, (3.4) is computable. Thus, τ t L BS t should bereplaced with (3.4). As a result, we suggest BS t plus (3.4) as an approximation of V t . Henceforth,we fix z t ∈ [0 , τ t ] arbitrarily. Before stating our approximation formula, we need to introduceadditional assumptions on the L´evy measure ν as follows: Assumption 3.4.
1. The L´evy measure ν is of the form ν ( dz ) = f ( z ) dz for z > . In addition,for any γ > , there exist C ν > depending on γ , and C ν ∈ (0 , (cid:15) ( T )) independent of γ suchthat f ( z ) ≤ C ν e − C ν z (3.5) for any z ≥ γ .2. For any t ∈ [0 , T ] , the distribution of X t is absolutely continuous with respect to the Lebesguemeasure. Remark 3.5.
Both examples introduced in Example 2.2: the IG-OU and the gamma-OU casessatisfy all the conditions in Assumption 3.4 from the view of Corollary 2.3 of [17].
Theorem 3.6.
Let (cid:101) V t := BS t + e − rτ t τ t (cid:90) ∞ z t ( K − e X t + ρz ) + { X t > log K } ν ( dz ) . Under Assumptions 2.1 and 3.4, there is a constant
C > such that (cid:12)(cid:12)(cid:12) V t − (cid:101) V t (cid:12)(cid:12)(cid:12) ≤ Cτ t for any t ∈ [0 , T ] , where C > is depending on X t , Σ t and T , and nondecreasing as a function of T , but independent of the choice of z t ∈ [0 , τ t ] . emark 3.7. From the view of Theorem 3.6, (cid:101) V t approximates to V t with order as τ t tendsto 0, that is, we can regard (cid:101) V t as an approximate option price. In particular, the second termin (cid:101) V t , namely (3.4) plays the role of a correction term. Now, we discuss its meaning. First ofall, we consider the case where the option is ITM, that is, X t > log K . Even if τ t is small, theITM option may change in a moment into an OTM one due to a big jump of the asset price,more precisely, a jump whose size is bigger than z := x − log K | ρ | . Thus, we can interpret that theintegrand in the correction term (3.4) eliminates the payoff of the ITM option when a big jumpoccurs, roughly speaking. In addition, since the probability that such a big jump occurs is nearlyequal to τ t (cid:82) ∞ z ν ( dz ) , the correction term is multiplied by τ t . On the other hand, since positive jumpsnever occur in the asset price process, we do not need to take into account of the reverse changes,that is, the changes of OTM options into ITM ones. Hence, the correction term takes the value of0 whenever X t ≤ log K . Remark 3.8.
Denote (cid:101) V t := BS t + e − rτ t τ t (cid:90) ∞ ( K − e X t + ρz ) + { X t > log K } ν ( dz ) . (3.6) It seems that (cid:101) V t is more natural as an approximation formula than (cid:101) V t with z t > from the viewof Remark 3.7. Note that (cid:101) V t coincides with (cid:101) V t when z t = 0 . However, as shown in Figure 1 below,the values of (cid:101) V t , represented by the red curve, are relatively large when the option is ITM, but nearto at-the-money (ATM), e.g., when < K < . . In such cases, z takes a small positivenumber. Thus, the integral in (3.6) takes a large value since ν ( dz ) diverges to ∞ as z tends to 0.Hence, we modify the correction term by taking a positive number as z t . As a matter of fact, theblue curve in Figure 1, representing the values of (cid:101) V t with z t = 0 . , approximates to the black curvesufficiently whenever the option is ITM. Figure 1: The values of V t , (cid:101) V t and (cid:101) V t versus strike price K from 440 to 470 at steps of 0.1 when T = 0 .
25 and t = 0 with parameter set introduced in Subsection 3.3. Note that S t = 468 .
44, and z t is set to 0.01. The black, blue and red curves represent the values of V t , (cid:101) V t and (cid:101) V t , respectively.9 .3 Numerical experiments We compute the values of (cid:101) V t numerically for the IG-OU case; and make sure that (cid:101) V t approximatesto V t sufficiently whenever the option is ITM. Recall that the L´evy measure of the IG-OU case isgiven as ν ( dz ) = λa √ π z − (1 + b z ) exp (cid:26) − b z (cid:27) (0 , ∞ ) ( z ) dz where a > b >
0. To simplify the notations, we denote c := λa √ π , c := λab √ π and c := b . Now, we illustrate how to compute (cid:101) V t . For α , β >
0, we define a function Γ( α, β ) asΓ( α, β ) := (cid:90) ∞ β e − x x α − dx, which is called the upper incomplete gamma function. DenotingΓ ( c, β ) := (cid:90) ∞ β e − cx x − dx = Γ (cid:0) , βc (cid:1) √ c and Γ ( c, β ) := (cid:90) ∞ β e − cx x − dx = 2 e − βc √ β − √ c Γ (cid:18) , βc (cid:19) for c > β >
0, we have (cid:90) ∞ z t ( K − e x z ) + ν ( dz )= (cid:90) ∞ z t ( K − e x + ρz ) (cid:16) c z − e − c z + c z − e − c z (cid:17) dz = Kc Γ ( c , z t ) + Kc Γ ( c , z t ) − e x c Γ ( c + | ρ | , z t ) − e x c Γ ( c + | ρ | , z t )whenever x > log K , where z = x − log K | ρ | and z t = z ∨ z t . Thus, we can compute (cid:101) V t numerically.Next, we introduce the numerical experiments implemented here. We fix t = 0, and set ρ = − . λ = 2 . a = 0 . b = 11 . r = 0 . S = 468 .
44 and Σ = 0 . z t is fixed to 0.01 for all cases.First, we compute the values of V , (cid:101) V , BS (0 , X , Σ ) and BS (0 , X , V ), where V is the averagesquared future volatility defined in Remark 3.3. Note that the values of V is computed by the fastFourier transform-based numerical scheme developed in Section 6 of [7] in order to compute thelocal risk-minimizing strategies for the BNS model as an extension of the so-called Carr-Madanmethod. Panel (a) in Figure 2 shows the values of V , (cid:101) V , BS (0 , X , Σ ) and BS (0 , X , V ) for thecall options with strike price K = 440 , . , . . . ,
480 when the maturity T is fixed to 0.25. In Panel(b), fixing K to 460, and moving T instead from 0.02 to 0.40 at steps of 0.02, we compute the same10alues for the option with K = 460, that is, the option is ITM, but not deep. Figure 2 indicatesthat (cid:101) V gives a very nice approximation of V for all ITM options. Moreover, it is clear from itsdefinition that (cid:101) V and BS (0 , X , Σ ) take the same value whenever the option is OTM, that is, S < K . Figure 2 also indicates that the difference between BS (0 , X , Σ ) and BS (0 , X , V ) arevery small as discussed in Remark 3.3.Panel (a) Panel (b)Figure 2: The values of V , (cid:101) V , BS (0 , X , Σ ) and BS (0 , X , V ) versus strike price K from 440 to480 at steps of 0.1 when T = 0 .
25 in Panel (a), and versus maturity T from 0.02 to 0.4 at stepsof 0.02 for the call option with strike price K = 460 in Panel (b). The black, blue, red and greencurves represent the values of V , (cid:101) V , BS (0 , X , Σ ) and BS (0 , X , V ), respectively.Next, we compute the approximation errors of (cid:101) V and BS (0 , X , Σ ), defined as | (cid:101) V − V | V and | BS (0 , X , Σ ) − V | V , respectively, under the same setting as Panel (a) of Figure 2 with various values of ρ . Panels (a)-(c)in Figure 3 indicate that, regardless of the value of ρ , the approximation errors of (cid:101) V are smallerthan those of BS (0 , X , Σ ) for all ITM options. Moreover, Panel (d) shows that the approximationerrors of (cid:101) V stay at a low level even though the time to maturity becomes longer. As a whole, theperformance of (cid:101) V as an approximation of V is sufficiently effective for any ITM option.11anel (a) Panel (b)Panel (c) ー C5 ヒ ゜゜ Time to Maturiy
Panel (d)Figure 3: The blue and red curves represent the approximation errors of (cid:101) V and BS (0 , X , Σ )respectively. Panels (a)-(c) are corresponding to the case where ρ = −
2, -4.7039 (the same valueas Figure 2), and -8 respectively. Note that other parameters take the same values as Panel (a) ofFigure 2. In Panel (d), ρ and K are fixed to -4.7039 and 460 respectively, but T is varying from0.02 to 0.4 at steps of 0.02. 12 Proof of Theorem 3.1
We shall show Theorem 3.1 by applying Ito’s formula twice to the Black-Scholes function.
Step 1.
Fix s, t ∈ [0 , T ) with s > t arbitrarily for the time being. Note that the function e − ru BS u , u ∈ [ s, t ] is sufficiently smooth to apply Ito’s formula. From the view of Lemma 4.2 below, we have e − rs BS s = e − rt BS t − r (cid:90) st e − ru BS u du + (cid:90) st e − ru ∂ t BS u du + (cid:90) st e − ru ∂ x BS u (cid:18) r + µ − Σ u (cid:19) du + (cid:90) st e − ru ( ∂ x BS u )Σ u dW u + 12 (cid:90) st e − ru ( ∂ x BS u )Σ u du + (cid:90) st e − ru ∂ σ BS u ( − λ Σ u ) du + (cid:90) st e − ru (cid:90) ∞ ∆ ρz,z BS u − N ( du, dz )= e − rt BS t + (cid:90) st e − ru D BS BS u du + (cid:90) st e − ru ∂ x BS u µdu + (cid:90) st e − ru ( ∂ x BS u )Σ u dW u + (cid:90) st e − ru ∂ σ BS u ( − λ Σ u ) du + (cid:90) st e − ru (cid:90) ∞ ∆ ρz,z BS u − N ( du, dz ) . (4.1)Now, we take the conditional expectation given X t and Σ t on both sides of (4.1). By (2.14) andLemmas 4.1 and 4.2, we have e − rs E [ BS s | X t , Σ t ] = e − rt BS t + E (cid:20)(cid:90) st e − ru ∂ x BS u µdu (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) + E (cid:20)(cid:90) st e − ru ∂ σ BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) + E (cid:20)(cid:90) st e − ru (cid:90) ∞ ∆ ρz,z BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) . (4.2)Taking the limitation on the left hand side as s tends to T , we havelim s → T E [ BS s | X t , Σ t ] = E [ BS T | X t , Σ t ] , since | BS s | ≤ sup t ∈ [0 ,T ] S t + K , which is integrable. Next, the partial derivatives ∂ x BS and ∂ σ BS are positive by (2.15) and (2.17). Thus, the monotone convergence theorem provides thatlim s → T E (cid:20)(cid:90) st e − ru ∂ x BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:34)(cid:90) Tt e − ru ∂ x BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) (4.3)and lim s → T E (cid:20)(cid:90) st e − ru ∂ σ BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:34)(cid:90) Tt e − ru ∂ σ BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) . s → T E (cid:20)(cid:90) st e − ru (cid:90) ∞ ∆ ρz,z BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:34)(cid:90) Tt e − ru (cid:90) ∞ ∆ ρz,z BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) . To summarize the above, taking the limitation on both sides of (4.2) as s tends to T , and multiplying e rt on both sides, we obtain V t = BS t + E (cid:34)(cid:90) Tt e − r ( u − t ) ∂ σ BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) + E (cid:34)(cid:90) Tt e − r ( u − t ) (cid:90) ∞ { ∆ ρz,z BS u + ∂ x BS u (1 − e ρz ) } ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) = BS t + I + I + E (cid:34)(cid:90) Tt e − r ( u − t ) L BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) , (4.4)since µ = (cid:82) ∞ (1 − e ρz ) ν ( dz ). Step 2.
We shall calculate the last term of (4.4). First of all, we fix t ∈ [0 , T ) arbitrarily, and define F ( u, x, σ ) := e − r ( u − t ) τ u L BS ( u, x, σ ) , u ∈ [ t, T ) . Lemma 4.3 ensures that, for any s, t ∈ [0 , T ) with t < s , L BS ( u, x, σ ) is a C , , -function on[ t, s ] × R × [ e − λT Σ , ∞ ). Remark that the domain of σ is restricted to [ e − λT Σ , ∞ ) from the viewof (2.4). Ito’s formula, together with (4.11) in Lemma 4.3, implies F ( s, X s , Σ s ) = F ( t, X t , Σ t ) − r (cid:90) st e − r ( u − t ) τ u L BS u du − (cid:90) st e − r ( u − t ) L BS u du + (cid:90) st e − r ( u − t ) τ u ∂ t L BS u du + (cid:90) st e − r ( u − t ) τ u ∂ x L BS u (cid:18) r + µ − Σ u (cid:19) du + (cid:90) st e − r ( u − t ) τ u ( ∂ x L BS u )Σ u dW u + 12 (cid:90) Tt e − r ( u − t ) τ u ( ∂ x L BS u )Σ u du + (cid:90) st e − r ( u − t ) τ u ∂ σ L BS u ( − λ Σ u ) du + (cid:90) st e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u − N ( du, dz )= F ( t, X t , Σ t ) − (cid:90) st e − r ( u − t ) L BS u du + (cid:90) st e − r ( u − t ) τ u ∂ x L BS u µdu + (cid:90) st e − r ( u − t ) τ u ( ∂ x L BS u )Σ u dW u + (cid:90) st e − r ( u − t ) τ u ∂ σ L BS u ( − λ Σ u ) du (cid:90) st e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u − N ( du, dz ) . (4.5)Remark that the above integral with respect to N ( du, dz ) is also well-defined by Lemma 4.5. Takingthe conditional expectation on both sides of (4.5), we have F ( s, X s , Σ s ) = F ( t, X t , Σ t ) − E (cid:20)(cid:90) st e − r ( u − t ) L BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) + E (cid:20)(cid:90) st e − r ( u − t ) τ u ∂ x L BS u µdu (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) + E (cid:20)(cid:90) st e − r ( u − t ) τ u ∂ σ L BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) + E (cid:20)(cid:90) st e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) (4.6)by Lemmas 4.4 and 4.5.Now, we take limits as s tends to T on both sides of (4.6). A similar argument with the proofof Lemma 4.2 yieldslim s → T E (cid:20)(cid:90) st e − ru (cid:90) ∞ ∆ ρz, BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:34)(cid:90) Tt e − ru (cid:90) ∞ ∆ ρz, BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) , from which, together with (4.3),lim s → T E (cid:20)(cid:90) st e − r ( u − t ) L BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:34)(cid:90) Tt e − r ( u − t ) L BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) holds. In addition, we havelim s → T E (cid:20)(cid:90) st e − r ( u − t ) τ u ∂ x L BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:34)(cid:90) Tt e − r ( u − t ) τ u ∂ x L BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) , and lim s → T E (cid:20)(cid:90) st e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:34)(cid:90) Tt e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) from the views of the proofs of Lemmas 4.4 and 4.5, respectively. Summarizing the above withLemmas 4.6 and 4.7, we obtain E (cid:34)(cid:90) Tt e − r ( u − t ) L BS u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) = F ( t, X t , Σ t ) + E (cid:34)(cid:90) Tt e − r ( u − t ) τ u ∂ x L BS u µdu (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) + E (cid:34)(cid:90) Tt e − r ( u − t ) τ u ∂ σ L BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) E (cid:34) (cid:90) Tt e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) . This completes the proof of Theorem 3.1. (cid:3)
Lemma 4.1. E (cid:20)(cid:90) st e − ru ( ∂ x BS u )Σ u dW u (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = 0 . (4.7) Proof.
Since (cid:98) S is a square integrable martingale, (cid:82) t (cid:98) S u Σ u dW u is also a square integrable martingale.Thus, (2.15) yields that E (cid:34)(cid:90) T e − ru ( ∂ x BS u ) Σ u du (cid:35) ≤ E (cid:34)(cid:90) T (cid:98) S u Σ u du (cid:35) < ∞ , which implies (4.7). (cid:3) Lemma 4.2.
The integral (cid:90) st e − ru (cid:90) ∞ ∆ ρz,z BS u − N ( du, dz ) is well-defined, and we have E (cid:20)(cid:90) st e − ru (cid:90) ∞ ∆ ρz,z BS u − N ( du, dz ) (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:20)(cid:90) st e − ru (cid:90) ∞ ∆ ρz,z BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) for any s, t ∈ [0 , T ) with t < s .Proof. From the view of Subsection 4.3.2 (p.231) of Applebaum [6], it suffices to see (cid:90) T (cid:90) ∞ E [ | ∆ ρz,z BS u | ] ν ( dz ) du < ∞ . Here, C denotes a positive constant, which may vary from line to line. For d ± and d ± ρz,z defined in(2.10) and (2.13) respectively, we have | d ± ρz,z − d ± | ≤ | x − log K + rτ t |√ τ t (cid:12)(cid:12)(cid:12)(cid:12) σ z − σ (cid:12)(cid:12)(cid:12)(cid:12) + | ρ | zσ z √ τ t + | σ z − σ |√ τ t ≤ | x − log K + rτ t |√ τ t | σ − σ z | σσ z + | ρ | zσ √ τ t + z √ τ t σ z + σ ) ≤ | x − log K + rτ t |√ τ t z σ + | ρ | zσ √ τ t + z √ τ t σ ≤ C (cid:18) | x | + 1 √ τ t + √ τ t (cid:19) zσ ∧ σ , (4.8)16here σ z is defined in (2.11). Now, (4.8) implies | ∆ ρz,z BS ( t, x, σ ) | = | e x z Φ( d + ρz,z ) − Ke − rτ t Φ( d − ρz,z ) − e x Φ( d + ) + Ke − rτ t Φ( d − ) |≤ e x z | Φ( d + ρz,z ) − Φ( d + ) | + e x | e ρz − | Φ( d + ) + Ke − rτ t | Φ( d − ρz,z ) − Φ( d − ) |≤ e x √ π | d + ρz,z − d + | + e x | ρ | z + Ke − rτ t √ π | d − ρz,z − d − | < C ( e x + 1) (cid:18) | x | + 1 √ τ t + √ τ t (cid:19) zσ ∧ σ + e x | ρ | z< C ( e x + 1)( | x | + 1) (cid:18) √ τ t + √ τ t + 1 (cid:19) z ∧ σ ∧ σ . Note that the second inequality is derived from | Φ( d + ρz,z ) − Φ( d + ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d + ρz,z d + φ ( ϑ ) dϑ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | d + ρz,z − d + |√ π , where φ is the probability density function of the standard normal distribution. Since the volatilityprocess Σ is bounded from below by (2.4), we have (cid:90) T (cid:90) ∞ E [ | ∆ ρz,z BS u | ] ν ( dz ) du ≤ C (cid:90) T (cid:18) √ τ u + √ τ u + 1 (cid:19) du (cid:90) ∞ zν ( dz ) (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) E (cid:32) sup t ∈ [0 ,T ] S t + 1 (cid:33) E (cid:32) sup t ∈ [0 ,T ] | X t | + 1 (cid:33) < ∞ (4.9)by (2.8) and (2.7), from which Lemma 4.2 follows. (cid:3) Lemma 4.3.
For any t , s ∈ [0 , T ) with t < s , and any partial derivative operator ∂ ∈{ ∂ t , ∂ x , ∂ x , ∂ σ } , ∂ L BS ( u, x, σ ) exists for ( u, x, σ ) ∈ [ t, s ] × R × [ e − λT Σ , ∞ ) , and we have ∂ L BS ( u, x, σ ) = L ∂BS ( u, x, σ ) . (4.10) In particular, D BS L BS ( u, x, σ ) = 0 (4.11) holds for ( u, x, σ ) ∈ [ t, s ] × R × [ e − λT Σ , ∞ ) .Proof. First of all, we show (4.10) for ∂ x . By the definition of L , (2.9) and (2.15), we have ∂ x L BS ( u, x, σ ) = ∂ x (cid:90) ∞ L z BS ( u, x, σ ) ν ( dz )= ∂ x (cid:90) ∞ (cid:26) e x z Φ( d + ρz ) − Ke − rτ u Φ( d − ρz ) − e x Φ( d + ) + Ke − rτ u Φ( d − )17 e x Φ( d + )(1 − e ρz ) (cid:27) ν ( dz )= ∂ x (cid:90) ∞ (cid:26) e x z (Φ( d + ρz ) − Φ( d + )) − Ke − rτ u (Φ( d − ρz ) − Φ( d − )) (cid:27) ν ( dz )= e x (1 + ∂ x ) (cid:90) ∞ e ρz (Φ( d + ρz ) − Φ( d + )) ν ( dz ) − Ke − rτ u ∂ x (cid:90) ∞ (Φ( d − ρz ) − Φ( d − )) ν ( dz ) . Remark that d ± and d ± ρz appeared in this proof are defined in (2.10) and (2.12) respectively, buttime parameter t is replaced with u . Note that | Φ( d + ρz ) − Φ( d + ) | ≤ | d + ρz − d + |√ π = 1 √ π | ρ | zσ √ τ u . Thus, | Φ( d + ρz ) − Φ( d + ) | is integrable with respect to ν ( dz ). Moreover, since φ (cid:48) is bounded, that is,there is a constant C φ (cid:48) > | φ (cid:48) ( d ) | < C φ (cid:48) (4.12)for any d ∈ R , we have | ∂ x (Φ( d + ρz ) − Φ( d + )) | = | ( ∂ x d + ρz ) φ ( d + ρz ) − ( ∂ x d + ) φ ( d + ) | = 1 σ √ τ u | φ ( d + ρz ) − φ ( d + ) | ≤ σ √ τ u C φ (cid:48) | ρ | zσ √ τ u , which is also integrable with respect to ν ( dz ). Similarly, we can see the integrability of | ∂ x (Φ( d − ρz ) − Φ( d − )) | . Thus, (4.10) holds when ∂ = ∂ x from the view of the dominated convergence theorem.As for ∂ x , we have ∂ x L BS ( u, x, σ ) = ∂ x L ∂ x BS ( u, x, σ )= ∂ x (cid:90) ∞ (cid:26) ∂ x BS ( u, x z , σ ) − ∂ x BS ( u, x, σ ) + ∂ x BS ( u, x, σ )(1 − e ρz ) (cid:27) ν ( dz )= ∂ x (cid:90) ∞ (cid:26) e x z Φ( d + ρz ) − e x Φ( d + ) + (cid:18) e x Φ( d + ) + e x σ √ τ u φ ( d + ) (cid:19) (1 − e ρz ) (cid:27) ν ( dz )= ∂ x (cid:90) ∞ (cid:26) e x z (Φ( d + ρz ) − Φ( d + )) + e x σ √ τ u φ ( d + )(1 − e ρz ) (cid:27) ν ( dz )= e x (1 + ∂ x ) (cid:90) ∞ (cid:26) e ρz (Φ( d + ρz ) − Φ( d + )) + 1 σ √ τ u φ ( d + )(1 − e ρz ) (cid:27) ν ( dz )by (2.16). Thus, we can show (4.10) for ∂ x by a similar argument with the case of ∂ x . Similarly,(4.10) holds for ∂ σ , since (2.17), together with (4.12), implies that (cid:12)(cid:12) ∂ σ L z BS ( u, x, σ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) √ τ u σ e x (cid:0) e ρz φ ( d + ρz ) − φ ( d + ) (cid:1) + √ τ u σ e x (cid:0) φ ( d + ) + ∂ x d + φ (cid:48) ( d + ) (cid:1) (1 − e ρz ) (cid:12)(cid:12)(cid:12)(cid:12) √ τ u σ e x (cid:26) e ρz | φ ( d + ρz ) − φ ( d + ) | + C φ (cid:48) σ √ τ u (1 − e ρz ) (cid:27) ≤ C φ (cid:48) e x σ ( e ρz | ρ | z + 1 − e ρz ) ≤ C φ (cid:48) e x e − λT Σ ( e ρz | ρ | z + 1 − e ρz ) , (4.13)which is integrable with respect to ν ( dz ). On the other hand, noting that ∂ t d ± = x − log K στ u − η ± σ √ τ u for u ∈ [ t, s ] ⊂ [0 , T ), where η ± is defined in (2.11), we can see (4.10) for ∂ t similarly.Summarizing the above, together with (2.14), we have (4.11). (cid:3) Lemma 4.4. E (cid:20)(cid:90) st e − r ( u − t ) τ u ( ∂ x L BS u )Σ u dW u (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = 0 for any s, t ∈ [0 , T ) with t < s .Proof. We show this lemma by the same way as the proof of Lemma 4.1. To this end, recall that ∂ x L BS ( u, x, σ ) = L ∂ x BS ( u, x, σ )= e x (cid:90) ∞ (cid:26) e ρz (cid:0) Φ( d + ρz ) − Φ( d + ) (cid:1) + φ ( d + ) σ √ τ u (1 − e ρz ) (cid:27) ν ( dz ) . Thus, we have (cid:12)(cid:12) ∂ x L BS ( u, x, σ ) (cid:12)(cid:12) ≤ e x πσ τ u (cid:26)(cid:90) ∞ ( e ρz | ρ | z + 1 − e ρz ) ν ( dz ) (cid:27) , (4.14)which implies E (cid:20)(cid:90) st e − r ( u − t ) τ u ( ∂ x L BS u ) Σ u du (cid:21) ≤ Ce rT T E (cid:20)(cid:90) st (cid:98) S u du (cid:21) ≤ Ce rT T E (cid:34) sup u ∈ [0 ,T ] | (cid:98) S u | (cid:35) < ∞ for some C >
0. This completes the proof of Lemma 4.4. (cid:3)
Lemma 4.5.
The integral (cid:90) st e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u − N ( du, dz ) is well-defined, and we have E (cid:20)(cid:90) st e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u − N ( du, dz ) (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:20)(cid:90) st e − r ( u − t ) τ u (cid:90) ∞ ∆ ρz,z L BS u ν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) for any s, t ∈ [0 , T ) with t < s . roof. By the same manner as Lemma 4.2, it suffices to see (cid:90) T τ u (cid:90) ∞ E [ | ∆ ρz,z L BS u | ] ν ( dz ) du < ∞ . (4.15)Recall that L BS ( t, x, σ ) = (cid:90) ∞ (cid:26) e x z Φ( d + ρz ) − Ke − rτ u Φ( d − ρz ) − e x Φ( d + ) + Ke − rτ u Φ( d − )+ e x Φ( d + )(1 − e ρz ) (cid:27) ν ( dz )= (cid:90) ∞ (cid:26) e x z (cid:0) Φ( d + ρz ) − Φ( d + ) (cid:1) − Ke − rτ u (cid:0) Φ( d − ρz ) − Φ( d − ) (cid:1) (cid:27) ν ( dz ) . (4.16)This implies∆ ρz,z L BS ( t, x, σ )= (cid:90) ∞ (cid:26) L w BS ( t, x z , σ z ) − L w BS ( t, x, σ ) (cid:27) ν ( dw )= (cid:90) ∞ (cid:26) e x z + w (cid:0) Φ( d + ρz + ρw,z ) − Φ( d + ρz,z ) (cid:1) − Ke − rτ u (cid:0) Φ( d − ρz + ρw,z ) − Φ( d − ρz,z ) (cid:1) − e x w (cid:0) Φ( d + ρw ) − Φ( d + ) (cid:1) + Ke − rτ u (cid:0) Φ( d − ρw ) − Φ( d − ) (cid:1) (cid:27) ν ( dw )= (cid:90) ∞ (cid:26) e x z + w (cid:90) d + ρz + ρw,z d + ρz,z φ ( ϑ ) dϑ − Ke − rτ u (cid:90) d − ρz + ρw,z d − ρz,z φ ( ϑ ) dϑ − e x w (cid:90) d + ρw d + φ ( ϑ ) dϑ + Ke − rτ u (cid:90) d − ρw d − φ ( ϑ ) dϑ (cid:27) ν ( dw )= ρσ z √ τ t (cid:90) ∞ (cid:90) w (cid:26) e x z + w φ ( d + ρz + ρζ,z ) − Ke − rτ u φ ( d − ρz + ρζ,z ) (cid:27) dζν ( dw ) − ρσ √ τ t (cid:90) ∞ (cid:90) w (cid:26) e x w φ ( d + ρζ ) − Ke − rτ u φ ( d − ρζ ) (cid:27) dζν ( dw )= ρσ z √ τ t (cid:90) ∞ (cid:90) w (cid:26) e x z + w φ ( d + ρz + ρζ,z ) − e x z + ζ φ ( d + ρz + ρζ,z ) (cid:27) dζν ( dw ) − ρσ √ τ t (cid:90) ∞ (cid:90) w (cid:26) e x w φ ( d + ρζ ) − e x ζ φ ( d + ρζ ) (cid:27) dζν ( dw )= ρe x √ τ t (cid:90) ∞ (cid:90) w ( e ρw − e ρζ ) (cid:26) e ρz σ z φ ( d + ρz + ρζ,z ) − σ φ ( d + ρζ ) (cid:27) dζν ( dw ) . (4.17)Note that the fifth equality of (4.17) comes from the following general fact: e x φ ( d + ) = Ke − rτ t φ ( d − )for any t ∈ [0 , T ), x ∈ R and σ >
0. In addition, the following inequality holds: (cid:12)(cid:12)(cid:12)(cid:12) e ρz σ z φ ( d + ρz + ρζ,z ) − σ φ ( d + ρζ ) (cid:12)(cid:12)(cid:12)(cid:12) φ ( d + ρz + ρζ,z ) (cid:12)(cid:12)(cid:12)(cid:12) e ρz σ z − σ (cid:12)(cid:12)(cid:12)(cid:12) + 1 σ | φ ( d + ρz + ρζ,z ) − φ ( d + ρζ ) |≤ √ π (cid:12)(cid:12)(cid:12)(cid:12) e ρz − σ z + 1 σ z − σ (cid:12)(cid:12)(cid:12)(cid:12) + C φ (cid:48) σ (cid:26) | d + ρz,z − d + | + | ρ | ζ √ τ t (cid:12)(cid:12)(cid:12)(cid:12) σ z − σ (cid:12)(cid:12)(cid:12)(cid:12) (cid:27) ≤ √ π (cid:18) | ρ | zσ + z σ (cid:19) + C φ (cid:48) σ (cid:26) C (cid:18) | x | + 1 √ τ t + √ τ t (cid:19) zσ ∧ σ + | ρ | ζ √ τ t z σ (cid:27) for some C >
0. Remark that C φ (cid:48) is the positive constant defined in (4.12), and the last inequalityis due to (4.8). Thus, (4.17) is less than Ce x ( | x | + 1) (cid:18) τ t + 1 √ τ t + 1 (cid:19) zσ ∧ σ (cid:90) ∞ ( w ∧ w ) ν ( dw )for some C >
0. As a result, substituting u , X u and Σ u for t , x and σ respectively, we can see(4.15) by a similar way with (4.9). (cid:3) Lemma 4.6. lim s → T F ( s, x, σ ) = 0 for any x ∈ R and σ > .Proof. First of all, we have τ s L BS ( s, x, σ ) = τ s (cid:90) ∞ (cid:110) e x z (cid:0) Φ( d + ρz ) − Φ( d + ) (cid:1) − Ke − rτ s (cid:0) Φ( d − ρz ) − Φ( d − ) (cid:1) (cid:111) ν ( dz ) . Now, we evaluate the above integrand as follows: τ s (cid:12)(cid:12)(cid:12) e x z (cid:0) Φ( d + ρz ) − Φ( d + ) (cid:1) − Ke − rτ s (cid:0) Φ( d − ρz ) − Φ( d − ) (cid:1) (cid:12)(cid:12)(cid:12) ≤ τ s (cid:110) e x z | ρ | z √ πσ √ τ s + K | ρ | z √ πσ √ τ s (cid:111) ≤ √ T | ρ | z √ πσ ( e x + K ) , which is integrable with respect to ν ( dz ). Thus, the dominated convergence theorem implieslim s → T F ( s, x, σ ) = (cid:90) ∞ lim s → T e − r ( s − t ) τ s L z BS ( s, x, σ ) ν ( dz ) = 0 . (cid:3) Lemma 4.7. lim s → T E (cid:20)(cid:90) st e − r ( u − t ) τ u ∂ σ L BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:21) = E (cid:34)(cid:90) Tt e − r ( u − t ) τ u ∂ σ L BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) . Proof.
By (4.13), we have | ∂ σ L BS ( u, x, σ ) | ≤ C e x σ (4.18)21or some C >
0. Thus, we can find a constant
C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) st e − r ( u − t ) τ u ∂ σ L BS u ( − λ Σ u ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ CT sup u ∈ [0 ,T ] S u , which is integrable with respect to P . Hence, Lemma 4.7 follows by the dominated convergencetheorem. (cid:3) Recall that V t − (cid:101) V t = I + I + I + I + I + τ t L BS t − e − rτ t τ t (cid:90) ∞ z t ( K − e X t + ρz ) + { X t > log K } ν ( dz ) . We shall show that each | I k | for k = 1 , . . . , Cτ t for some C >
C > (cid:12)(cid:12)(cid:12)(cid:12) L BS ( t, x, σ ) − (cid:90) ∞ z t ( K − e x z ) + { x> log K } ν ( dz ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ τ t (5.1)for ( t, x, σ ) ∈ [0 , T ) × R × [ e − λT Σ , ∞ ). Remark that all positive constants C > X t (or x ), Σ t (or σ ) and T , and nondecreasing as a function of T ,but independent of the choice of z t ∈ [0 , τ t ). Without loss of generality, we may assume that ρ < ρ = 0. Step 1.
Firstly, we see | I | ≤ Cτ t for some C >
0. Since we have e x φ ( d + ) = Ke − rτ u φ ( d − ) , (2.17) implies (cid:12)(cid:12) ∂ σ BS ( u, x, σ ) (cid:12)(cid:12) = √ τ u σ Ke − rτ u φ ( d − ) ≤ √ τ u σ Ke − rτ u √ π . (5.2)This, together with (2.4) and (2.5), provides | I | ≤ λ √ τ t K √ πe − λT/ Σ E (cid:34)(cid:90) Tt Σ u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) ≤ λ √ τ t K √ πe − λT/ Σ (cid:18) τ t Σ t + τ t (cid:90) ∞ zν ( dz ) (cid:19) < Cτ t for some C >
0, since (cid:15) ( t ) ≤ t holds for any t ∈ [0 , T ]. Remark that C is depending on Σ t , andnondecreasing as a function of T .Next, we show | I | ≤ Cτ t for some C >
0. (5.2) implies that | (∆ ρz,z − ∆ ρz, ) BS ( u, x, σ ) | = | BS ( u, x z , σ z ) − BS ( u, x z , σ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) σ z σ ∂ σ BS ( u, x z , ˜ σ ) d ˜ σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup σ ≤ ˜ σ ≤ σ z | ∂ σ BS ( u, x z , ˜ σ ) | z ≤ √ τ u √ πσ Ke − rτ u z. As a result, we obtain by (2.4) | I | ≤ E (cid:34)(cid:90) Tt (cid:90) ∞ √ τ u √ πe − λT/ Σ Ke − rτ u zν ( dz ) du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) ≤ Cτ t for some C > I , (4.14) ensures that there is a constant C > | ∂ x L BS ( u, x, σ ) | ≤ C e x σ √ τ u , which implies that | I | ≤ Cτ t by (2.4) and (2.8). To see the same evaluation for I , (4.18) impliesthat | I | ≤ C E (cid:34)(cid:90) Tt e − r ( u − t ) τ u u e X u λ Σ u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) ≤ C E (cid:34)(cid:90) Tt τ u S u du (cid:12)(cid:12)(cid:12) X t , Σ t (cid:35) ≤ C E (cid:34) sup u ∈ [0 ,T ] S u (cid:35) τ t √ T for some C >
0, which is depending on neither X t , Σ t nor T .Next, we evaluate | I | . From the views of (4.14) and (4.18), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ ∆ ρz,z L BS ( t, x, σ ) ν ( dz ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) ∞ (cid:8) |L BS ( t, x z , σ z ) − L BS ( t, x, σ z ) | + |L BS ( t, x, σ z ) − L BS ( t, x, σ ) | (cid:9) ν ( dz ) ≤ (cid:90) ∞ (cid:40) | ρ | z sup x z ≤ y ≤ x (cid:12)(cid:12) ∂ x L BS ( t, y, σ z ) (cid:12)(cid:12) + z sup σ ≤ ˜ σ ≤ σ z (cid:12)(cid:12) ∂ σ L BS ( t, x, ˜ σ ) (cid:12)(cid:12)(cid:41) ν ( dz ) ≤ Ce x (cid:90) ∞ (cid:40) | ρ | z √ πσ z √ τ t + z σ (cid:41) ν ( dz ) ≤ Ce x σ ∧ σ (cid:18) √ τ t + 1 (cid:19) , where C > x , σ and T . By (2.4), we obtain | I | ≤ Cτ t for some C > Step 2.
The aim of this step is to show (5.1). First of all, we prove that there exists a constant
C > (cid:12)(cid:12) L BS ( t, x, σ ) (cid:12)(cid:12) ≤ C √ τ t , (5.3)when x < log K . Note that we do not need to consider the case of x = log K , since P ( X t = log K ) =0 holds by the condition 2 of Assumption 3.4. Since there exists a constant C > (cid:12)(cid:12) L BS ( t, x, σ ) (cid:12)(cid:12) ≤ C √ τ t τ t > τ t . Now, we fix arbitrarily τ t > x − log K + η + τ t < . We have then d ± ρz ≤ d ± < z >
0, and we can find a constant
C > d ± ) − Φ( d ± ρz ) ≤ | ρ | zσ √ τ t φ ( d ± ) ≤ | ρ | zσ √ τ t φ ( d + ) ≤ | ρ | z √ πσ √ τ t exp (cid:26) − ( x − log K ) σ τ t (cid:27) ≤ (cid:114) π | ρ | zσ √ τ t ( x − log K ) e (5.4)for any z >
0, since βe − αβ ≤ αe holds for any α , β >
0. Thus, together with (4.16), we have |L BS ( t, x, σ ) | ≤ e x (cid:90) ∞ (cid:0) Φ( d + ) − Φ( d + ρz ) (cid:1) ν ( dz ) + K (cid:90) ∞ (cid:0) Φ( d − ) − Φ( d − ρz ) (cid:1) ν ( dz ) ≤ C √ τ t for some C > x > log K . To this end, we decompose the left hand side of(5.1) into the following three terms: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) z L z ( t, x, σ ) ν ( dz ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) z tz L z ( t, x, σ ) ν ( dz ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ∞ z t ( L z ( t, x, σ ) − ( K − e x z )) ν ( dz ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5.5)where z := x − log K | ρ | and z t := z ∨ z t . In the rest of this proof, we fix τ t > x − log K η − τ t > C √ τ t for some C > < d − ρz for any z ∈ (0 , z ], we can find a constant C > φ ( d ± ρz ) σ √ τ t ≤ √ πσ √ τ t exp (cid:26) − ( x − log K + ρz + η ± τ t ) σ τ t (cid:27) ≤ √ πσ √ τ t exp − (cid:16) x − log K + η − τ t (cid:17) σ τ t ≤ √ πσ √ τ t exp (cid:26) − ( x − log K ) σ τ t (cid:27) ≤ C √ τ t for any z ∈ (0 , z ] by (5.6) and (5.4). Thus, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) z L z ( t, x, σ ) ν ( dz ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) z e x z (Φ( d + ) − Φ( d + ρz )) ν ( dz ) + K (cid:90) z (Φ( d − ) − Φ( d − ρz )) ν ( dz )24 (cid:90) z e x φ ( d + ρz ) | ρ | zσ √ τ t ν ( dz ) + K (cid:90) z φ ( d − ρz ) | ρ | zσ √ τ t ν ( dz ) ≤ C ( e x + K ) | ρ |√ τ t (cid:90) z zν ( dz ) , which guarantees the existence of C > C √ τ t .Secondly, we prove that the second term of (5.5) has the same evaluation as the first term.Remark that we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) z tz L z ( t, x, σ ) ν ( dz ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) z tz e x z (Φ( d + ) − Φ( d + ρz )) ν ( dz ) + K (cid:90) z tz (Φ( d − ) − Φ( d − ρz )) ν ( dz )= (cid:90) z tz e x z (cid:90) d + d + ρz φ ( ϑ ) dϑν ( dz ) + K (cid:90) z tz (cid:90) d − d − ρz φ ( ϑ ) dϑν ( dz ) . (5.7)To give an evaluation of the second term of (5.7), we denote h ( ϑ ) := z + η − τ t − σ √ τ t ϑ | ρ | . By the condition (3.5), we can find C ν > C ν > ν ( dz ) ≤ C ν e − C ν z dz ≤ C ν dz for any z ≥ z , from which we have (cid:90) z tz (cid:90) d − d − ρz φ ( ϑ ) dϑν ( dz )= (cid:90) d − d − t (cid:90) z tz ∨ h ( ϑ ) ν ( dz ) φ ( ϑ ) dϑ = (cid:90) d − − x − log K σ √ τt d − t (cid:90) z t h ( ϑ ) ν ( dz ) φ ( ϑ ) dϑ + (cid:90) d − d − − x − log K σ √ τt (cid:90) z tz ν ( dz ) φ ( ϑ ) dϑ ≤ C ν (cid:32) (cid:90) d − − x − log K σ √ τt d − t ( z t − h ( ϑ )) φ ( ϑ ) dϑ + z t (cid:90) d − d − − x − log K σ √ τt φ ( ϑ ) dϑ (cid:33) ≤ C ν (cid:32) (cid:90) d − − x − log K σ √ τt d − t ( | z t − z | + | z − h ( ϑ ) | ) φ ( ϑ ) dϑ + ( z + τ t ) (cid:90) d − d − − x − log K σ √ τt φ ( ϑ ) dϑ (cid:33) ≤ C ν (cid:40) (cid:90) ∞−∞ (cid:18) τ t + | η − | τ t + σ √ τ t | ϑ || ρ | (cid:19) φ ( ϑ ) dϑ + ( z + τ t ) (cid:90) d − d − − x − log K σ √ τt φ ( ϑ ) dϑ (cid:41) , where d − t := d − ρz t . As for the integral of the second term, the same sort argument as (5.4) implies (cid:90) d − d − − x − log K σ √ τt φ ( ϑ ) dϑ ≤ φ (cid:18) x − log K σ √ τ t (cid:19) x − log K σ √ τ t ≤ C √ τ t C >
0, which is independent of the choice of z t . Similarly, we obtain the same evaluationfor the first term of (5.7). Hence, the second term of (5.5) is less than C √ τ t for some C > z t = 0, inother words, z t = z . Noting that 1 − Φ( ϑ ) ≤ √ πϑ for any ϑ >
0, and x − log K + η ± τ t > x − log K , we have 1 − Φ( d ± ) ≤ √ πd ± < (cid:114) π σ √ τ t x − log K , which provides that, for any z > z , |L z ( t, x, σ ) − ( K − e x z ) | = | e x z (cid:0) Φ( d + ρz ) − Φ( d + ) + 1 (cid:1) − Ke − rτ u (cid:0) Φ( d − ρz ) − Φ( d − ) + 1 (cid:1) + K ( e − rτ t − |≤ e x z Φ( d + ρz ) + ( e x z + K ) (cid:114) π σ √ τ t x − log K + K Φ( d − ρz ) + Krτ t . Thus, since ν ([ z , ∞ )) < ∞ and Φ( d + ρz ) ≥ Φ( d − ρz ), it suffices to see that (cid:90) ∞ z Φ( d + ρz ) ν ( dz ) < C √ τ t for some C > (cid:90) ∞ z Φ( d + ρz ) ν ( dz ) = (cid:90) ∞ z (cid:90) d + ρz −∞ φ ( ϑ ) dϑν ( dz ) = (cid:90) η + σ √ τ t −∞ (cid:90) z + η + τt − σ √ τtϑ | ρ | z ν ( dz ) φ ( ϑ ) dϑ ≤ (cid:90) η + σ √ τ t −∞ (cid:90) z + η + τt − σ √ τtϑ | ρ | z C ν e − C ν z dzφ ( ϑ ) dϑ ≤ C ν e − C ν z C ν (cid:90) η + σ √ τ t −∞ (cid:18) − exp (cid:26) − C ν η + τ t − σ √ τ t ϑ | ρ | (cid:27)(cid:19) φ ( ϑ ) dϑ ≤ C ν e − C ν z C ν (cid:90) η + σ √ τ t −∞ C ν η + τ t − σ √ τ t ϑ | ρ | φ ( ϑ ) dϑ ≤ C ν e − C ν z | ρ | η + τ t σ √ τ t √ π + (cid:90) η + σ √ τ t (cid:0) η + τ t − σ √ τ t ϑ (cid:1) φ ( ϑ ) dϑ < C √ τ t for some C >
0. This completes the proof of Theorem 3.6. (cid:3) Conclusions
An Al`os type decomposition formula for the vanilla call option for the BNS model has been derivedby using Ito’s formula twice, and an approximation option pricing formula also has been provided.Moreover, numerical results introduced in Subsection 3.3 indicate that our approximation (cid:101) V t iseffective whenever the option is ITM. Note that the development of an approximation formulawhich is effective for OTM options is still open. Besides, the obtained approximation formulawould enable us to develop an approximation of implied volatilities and a calibration method formodel parameters, but we leave them to future works. Acknowledgments
Takuji Arai gratefully acknowledges the financial support of the MEXT Grant in Aid for ScientificResearch (C) No.18K03422.